Shadow tomography is the task of simultaneously estimating the outcome probabilities associated with \(M\)  2-outcome measurements up to accuracy \(\epsilon \) : \(p_i (\rho ) = \mathrm {tr}(E_i \rho )\) , where each \(E_i\)  is a positive semi-definite matrix with operator norm at most one [1]}, [2]}, [3]}, [4]}.
The best existing result is given by [4]}. They showed that
\(N= \mathcal {O} \left(\log (M)^2 \log (d) /\epsilon ^4 \right)\)  copies of the unknown state suffice to achieve this task.
Their protocol is based on an improved quantum threshold search: finding an observable \(E_i\)  with the expectation value \(\operatorname{tr}(E_i \rho )\)  exceeding a certain threshold. Then, they combine this with online learning of quantum states similar Aaronson's original protocol. [1]}.
A more experimentally friendly shadow tomography protocol has been proposed in [7]}. It only yields competitive scaling complexities for a restricted set of observables, but can
implemented on state-of-the-art quantum platforms [8]}, [9]}.
